Function codomain
Given a
function f:
A →
B, the
set B is called the
codomain of
f.
The codomain is not to be confused with the
range f(
A), which is in general only a
subset of
B.
Let the function f be a function on the real numbers:
- f: R → R
defined by
- f: x → x^{2}
The codomain of
f is
R, but clearly
f(
x) never takes
negative values, and thus the range is in fact the set
R^{+} -- non-negative reals, ie the
interval [0,∞):
- 0 ≤ f(x) < ∞
One could have defined the function
g thus:
- g: R → R^{+}
- g: x → x^{2}
While
f and
g have the same effect on a given number, they are not, in the modern view, the same function since they have different codomains.
The codomain can affect whether or not the function is a surjection; in our example, g is a surjection while f is not.
See also: Function domain, Function range